# A Multi-Value Sequence Generated by Power Residue Symbol and Trace Function over Odd Characteristic Field

## Abstract

<p>This paper proposes a new multi-value sequence generated by utilizing primitive element, trace, and power residue symbol over odd characteristic finite field. In detail, let <i>p</i> and <i>k</i> be an odd prime number as the characteristic and a prime factor of <i>p</i>-1, respectively. Our proposal generates <i>k</i>-value sequence <i>T</i>={<i>t<sub>i</sub></i> | <i>t<sub>i</sub></i>=<i>f<sub>k</sub></i>(Tr(ω<i><sup>i</sup></i>)+<i>A</i>)}, where ω is a primitive element in the extension field $\F{p}{m}$, Tr(⋅) is the trace function that maps $\F{p}{m} \rightarrow \f{p}$, <i>A</i> is a non-zero scalar in the prime field $\f{p}$, and <i>f<sub>k</sub></i>(⋅) is a certain mapping function based on <i>k</i>-th power residue symbol. Thus, the proposed sequence has four parameters as <i>p</i>, <i>m</i>, <i>k</i>, and <i>A</i>. Then, this paper theoretically shows its period, autocorrelation, and cross-correlation. In addition, this paper discusses its linear complexity based on experimental results. Then, these features of the proposed sequence are observed with some examples.</p>

## Journal

• IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E99.A(12), 2226-2237, 2016

The Institute of Electronics, Information and Communication Engineers

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